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Parikh vectors
8QBK_1 1GGI_1 5PSM_1 Letter Amino acid
6 5 1 C Cysteine
5 3 10 H Histidine
17 12 4 N Asparagine
19 16 13 D Aspartic acid
37 15 18 L Leucine
25 30 7 S Serine
13 10 8 Q Glutamine
7 12 6 P Proline
1 3 0 W Tryptophan
17 11 10 V Valine
5 1 7 M Methionine
18 8 6 F Phenylalanine
16 11 11 A Alanine
22 9 14 R Arginine
15 10 13 E Glutamic acid
13 13 7 G Glycine
23 11 5 I Isoleucine
28 11 6 K Lycine
9 19 5 T Threonine
11 8 5 Y Tyrosine 

8QBK_1|Chains A[auth E], B[auth F], O[auth G], T|Retron Ec86 putative ribosyltransferase/DNA-binding protein|Escherichia coli BL21(DE3) (469008)
>1GGI_1|Chains A[auth L], D[auth M]|IGG2A 50.1 FAB (LIGHT CHAIN)|Mus musculus (10090)
>5PSM_1|Chains A, B|Bromodomain-containing protein 1|Homo sapiens (9606)
Protein code \(c\) LZ-complexity \(\mathrm{LZ}(w)\) Length \(n=|w|\) \(\frac{\mathrm{LZ}(w)}{n /\log_{20} n}\) \(p_w(1)\) \(p_w(2)\) \(p_w(3)\) Sequence \(w=f(c)\)
8QBK , Knot 134 307 0.83 40 180 295 
MNKKFTDEQQQQLIGHLTKKGFYRGANIKITIFLCGGDVANHQSWRHQLSQFLAKFSDVDIFYPEDLFDDLLAGQGQHSLLSLENILAEAVDVIILFPESPGSFTELGAFSNNENLRRKLICIQDAKFKSKRSFINYGPVRLLRKFNSKSVLRCSSNELKEMCDSSIDVARKLRLYKKLMASIKKVRKENKVSKDIGNILYAERFLLPCIYLLDSVNYRTLCELAFKAIKQDDVLSKIIVRSVVSRLINERKILQMTDGYQVTALGASYVRSVFDRKTLDRLRLEIMNFENRRKSTFNYDKIPYAHP
1GGI , Knot 103 218 0.84 40 152 213 
DIVLTQSPGSLAVSLGQRATISCRASESVDDDGNSFLHWYQQKPGQPPKLLIYRSSNLISGIPDRFSGSGSRTDFTLTINPVEADDVATYYCQQSNEDPLTFGAGTKLEIKRADAAPTVSIFPPSSEQLTSGGASVVCFLNNFYPKDINVKWKIDGSERQNGVLNSWTDQDSKDSTYSMSSTLTLTKDEYERHNSYTCEATHKTSTSPIVKSFNRNEC
5PSM , Knot 74 156 0.79 38 120 151 
MHHHHHHSSGVDLGTENLYFQSMEQVAMELRLTELTRLLRSVLDQLQDKDPARIFAQPVSLKEVPDYLDHIKHPMDFATMRKRLEAQGYKNLHEFEEDFDLIIDNCMKYNARDTVFYRAAVRLRDQGGVVLRQARREVDSIGLEEASGMHLPERPA

Let \(P_w(n)\) be the set of distinct subwords (intervals) in a word \(w\). Let \(p_w(n)\) be the cardinality of \(P_w(n)\). Let \(f(c)\) be the sequence in FASTA with 4-symbol Protein Data Bank code \(c\).

\(|P_{f(8QBK_1)}(2) \setminus P_{f(1GGI_1)}(2)|=91\), \(|P_{f(1GGI_1)}(2) \setminus P_{f(8QBK_1)}(2)|=63\). Let \( Z_k(x,y)=|P_x(k)\setminus P_y(k)|+|P_y(k)\setminus P_x(k)| \) be a LZ76 style (set of subwords) Jaccard distance numerator for \(P(k)\).Hydrophobic-polar version of Sequence 1:1000100000001110100011001101010111011011000010001001110100101101001100111101000110100111011011111100110100111100000100011010010100000110011101100100001100000010010000101100101000111010010000010001101101001111010110010000100111011000011001110011001100001101001001011110010011000010010101101000000010000110101
Pair \(Z_2\) Length of longest common subsequence
8QBK_1,1GGI_1 154 4
8QBK_1,5PSM_1 168 4
1GGI_1,5PSM_1 172 3

Newick tree

 
[
	5PSM_1:87.51,
	[
		8QBK_1:77,1GGI_1:77
	]:10.51
]

Let d be the Otu--Sayood distance d.
Let d1 be the Otu--Sayood distance d1. (This makes the 4TYN sequence AAAAAA a close match...)
A roughly speaking expected distance is \((0.85)(0.8)(\frac{525 }{\log_{20} 525}-\frac{218}{\log_{20}218})=88.2\)
Status Protein1 Protein2 d d1/2
Query variables 8QBK_1 1GGI_1 112 95
Was not able to put for d
Was not able to put for d1

In notation analogous to [Theorem 16, Kjos-Hanssen, Niraula and Yoon (2022)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]